J. Korean Math. Soc.
V01.9. No. 1. 1972
SUBMANIFOLDS OF CODIMENSION 2 OR 3 WITH PARALLEL SECOND FUNDAMENTAL TENSOR
By KENTARO YANO AND SHIGERU ISHIHARA
1
§O. Introduction
In a previous paper [4J, the present authors have generalized formulas of Nomizu and 'Smyth [l] to the case of submanifolds of higher codimension and used them to study
"Submanifolds, immersed in a space of constant curvature, whose normal bundle is locally parallelizable and mean curvature vector field is parallel in the normal bundle.
The main purpose of the present paper is to study submanifolds of codimension 2 or3, immersed in a space of constant curvature, whose second fundamental tensor is parallel with respect to van der Waerden-Bortolotti covariant differentiation. In § I, we recall fundamental formulas of the isometric immersion and, in § 2, prove some lemmas con- -cerning submaniIolds, immersed in a space of constant curvature, whose second funda- mental tensor is parallel and whose holonomy group of the connection induced in the normal bundle is abelian. In § 3 and § 4, we prove some propositions concerning sub- manifolds of codimension 2 or 3, immersed in a space of constant curvature, whose second fundamental tensor is parallel.
§1. Submanifolds
Let there be given an n-dimensional differentiable manifold!) M of classC'"covered by .a system X of coordinate neighborhoods {V; y"}2) and immersed as a submanifold in an m-dimensional Riemannian manifold {M,g} with metric tensor g. The immersion will be denoted by i : M-+M (n>2, r=m-n~1), that is, M is a submaniIold of codimension r in (M, g) with immersion i: M-+M. The enveloping manifold
1\1
is assumed to be -covered by a system ~ of coordinate neighborhoods {O; xh}ID. Thus we may assume that, for any element U of X, i(V) is contained in some element0
of ~. The immer-"Sion i: M-+M is locally expressed by (1.1)
in i(V)(cO), where :respect to Y·, we put
Xh=Xh(y")
to;
xh}E~ and {V; y.}EJ'. Differentiating (1.1) partially with{I. 2) Bhh=Ohxh, Ob=O/oyb
in i(V). Then, for each fixed index b, we have along i(V) a local tangent vector field Received by the editors Feb. 19. 1972.
1) Manifolds. mappings and geometric objects we discuss are assumed tobedifferentiable and of class CO'.
2) The indices a. b, c. d, erun over the range {I, 2, ''', n} and the summation convention will be used with respect to this system of indices.
.3) The indices h.i,j,k run over the range {I. 2, "',m} and the summation convention will be used with respect to this system of indices.
2 Kentaro Yano and Shigeru Ishihara
B" to i(M) with components B"k. We consider overi(U) r(=m-n) orthonormal locaf cross-sections C,4'1 of the normal hundle N(M) of the submanifold M anddenote byC/.
components of C, with respect to {V; xi}. where here and in the sequel we identify i(M) with M and i(U) with U for any UEI.
The induced metric g of the submanifold M has components ge"=gjjB/B,,jin tU;y"}, where gji are components of g in {V; xh}. The induced metric g of N(M) has comp- onents g,.=gjzC/C,/ in
tu;
y"} with respect to the orthonormal frame {C,}, where g,,,, is numerically equal toa,,,,.
H we put(.8"j, C·i )=(B.h, C/)-l, then we find J3ai= B.hgkghi and C·i=C,hg'·ghj, where (gcb)=(gc,,)-l and (g'.)=(g,,,,)-l.We denote by f'cB. and f'ec,local vector fields" along U, respectively with components.
fJeB.k=oeB.h+ Vi} B/B/, f'eC,h=OeC,k+ {jki} B/C,i
with respect to
to;
xi}, where {jhi} are the Christoffel symbols formed with gjio Then.the tangential components (fJeB.Y of f'eBb is given by
(1.3) (VeB.)T
=
{ca.} Ba,{ca.} being the Christoffel symbols formed with gc.,and the normal components (f'cC,y- of f'e
c
7 by(1.4) (VeC,)"J.=Fc",c,c>
where Fc"', denote components of the connection 17induced inN(M) with respect to the·
frame {C,} and
(1.5) Fe"',
+
Fe'",=
O.The normal components(VeB"Yof f'eBh and the tangential component(VeC,)TofVeC,.
have respectively components of the form
(1.6) fTeB.h=oeBhh+ {j\}B/B.i- {ca"}Bah=he,,·C,,,\
(1.7) fTeC,h=oeCl+ {jh;}B/C';-F?,c,.A=-heayB"k
in
tu;
xi'}, where heh·are components of the second fundamental tensor of the subma- nifold M intu;
yo} with respect to the normal frame {Cy} and hcay=heh"'gh"g",y" The he.'" is symmetric, i. e., he"z=hhcz•Hwe put
O·
8) H=ng<"ke"·C",,1then we see that H is a global cross-section in N(M), which is called the mean curva- ture vectorof the submanifold M. The length I!HII ofH is called the mean curvature of the submanifold M. When the mean curvature IIHII vanishes identicallyin M, the.
submanifold M is called a minimal srihmanifold.
Now, we assume that the mean curvature vector H vanishes nowhere in M. Then we can define a unit cross-section D=H/IIHII in N(M) and hence, putting D=DzC"" a.
global tensor field of type (0, 2) with components
4) The indices x, y. z run over the range {m+l• •••, n} and the summation convention willhe used also with respect to thissystem of indices.
Submanifolds of codimension 2 or 3 with parallel second fundamental tensor. ::I
(1.g)
in M. The tensor field hcb IS symmetric, i. e.,hcb=hbc' When the condition (1.10)
hc,,'=gebAX
being a non-zero cross-section in N(M), the submanifold M is When heb' vanishes identically in M, the submanifold M is said to is satisfied, the submanifold M is said to be pseudo-umbilical. When the condition (1. 11)
is satisfied, A=AxC, said to be umbilical.
be totally geodesic.
We denote by Kkjihcomponents of the curvature tensor of (M,g) in {a; x h} and by K dcb athose of the curvature tensor of (M, g) in {U; y"}. We denote by Kde/ compo- nents of the curvature tensor of the connection j7 induced in N(M) with respect to the normal frame {Cy} , K dey' being defined by
(1.12) Kdeyx=OdT/y-oeTdXy+Td'.T/y-T/.Tdzy,
Then, differentiating (1.6) and (1. 7) covariantly and taking account of the Ricci iden- tities, we have the structure equations of the submanifold M immersed in (M,g) as follows:
(1.13) (1.14) (1.15)
where we have put
and (1.16)
Kdcba=Kdc"a
+
(hdaxhe"x - he"xhd"x), 0= Kdebx+er
dhe"x -er
ehd"x,KdCba=KkjihBiBeiB/Bah, Kde"x=KkiihB/B/BbiCxh>
Kde/=KkiihB/B/C/Cxh
If the enveloping manifold (M,g) is of constant curvature e, i.e., if Kkiih=C(Okhgii-olgki),
then we have, substituting the equation above in (1.13), (1.14) and Cl.15) respectively, (1.17)
(1.18) (1.19)
Transvecting O.17) with geb and contracting the indices a and d, we find K=n(n-1)e
+
n211HI12 - hebW"x'where K=ge"Kacba is the scalar curvature of (M, g) and hcbx=hd/gdcgobgxY' Thus, using the equation above, we have (cf. [3J)
LEMMA1.1. Let M be a submanifold immersed in a Riemannian manifold of constant
4 Kentaro Yano and ShigeruIshihara
euroaellTe c. Tlum, if tke scalar CUT'Dflt1Ire K of (M, g) with induced metric g satisfies IC:;:n(n-1)c, (dim M=n), and ifM is a minitlUll submanifold, then M is totally geodesil:.
A Riemannian manifold (M, g) is said to be irreducihle, if itslinearholonomy group is irreducible as a linear group. A Riemannjan manifold is said to be locally irreducihle if any open submanifold of (M, g) is irreducible as a Riemannian manifold having the restriction of g as metric tensor. A 8ubmanifold M is said to be (locally) irreducihle, if the Riemannjan manifoid (M, g) with induced metric g is Oocally) irreducible.
H the second fundamental tensor h.." satisfies the condition
(l. 20) v,/I,••"=O,
then we say that the submanifoldM has parallelsecond fimdamental teRSDr. The Ricci identity and (1. 20) imply
(l.21) - Ke",."ku"-~."hca
"+
K.1tIY"h••Y=O.Thus, if V",h••"=O, then (1.21) holds.
We assume that the submanifold M has parallel second fundamental tensor. Then, using (1. 5), (l. 8), (1. 9) and (1. 20), we find
(Tdh••=O,/l,..-
Lt".}
hab - {",".}k.,,=O.Thus, if the submanifold M is, moreover, irreducible, then we find h••=ag••, a being a constant, because of hc.=k... Consequently, we have
PROPOSITION 1. 1. Let M be a mbmanifold with parallel second fundamental tensor.
If M is irreducilJle and not a minitlUll S1JlJmanifold. then it is a pseudo-umbilicalmh- manifoldwith cOtl#ant mean CUT'Dflt1Ire.
We de1ine the covarlant derivative of the curvature tensor of the connectionl7 induced in H(M) as a tensor field with components
(l.22)
Then, diHerentiating (1.17) and (l.19) covariantlyand taking account of (1.22), we have
LEMMA 1.2. LetM be a mhmanifold immersedin a Riemannianmanifold ofcOlUtant euroatIITe and assume that M has parallel set:oIUl,ftuIJamentaltensor. Tlum
(l. 23) V$.7'''=0
holds tztUl tke Riemannian manifold (M, g) with eke induced metric g is locally gym.
metrie. i.
e..
v.,K.,.,,"=O, eke #alar ~ K of (M, g) being eemstant.H a submanifold M immersed in a Riemannian manifold of constant curvature has parallel second fundamental tensor, then, by Lemma L2, (M, g) is locally symmetric, If being the induced metric of M. Thus, if such a submanifold M is irreducible, then M is loca1ly irreducible.
Submanifolds of codimension 2 or 3 with parallel second fundamental tensor. 5
§2. Normal bundles with abelian holonomy group
Let there be given a submanifold M of codimension r of (M, g). We assume that the holonomy group I/) of the connection i1 induced in the normal bundle N(M) is abelian.
Then I/) is similar to a subgroup of the linear group 1Jf of all matrices of the form (2.1)
0< 2s:S;;r.
in the orthogonal group OCr). where R(2) denotes the group of rotations and Ao a diagonal (r-s, r-s)-matrix with diagonal elements +1 or -1. Thus the Lie algebra of the holonomy group I/) is a subalgebra of the Lie algebra of all matrices of the form
(2.2)
a=1.···.s.
(2.3)
where Or-. denotes the zero (r-s. r-s)-matrix. Therefore there is. along each coor- dinate neighborhood U of M. orthonormal cross-sections Cy of N(M) satisfying the following conditions (A) and (B):
(A) The connection i7 induced in N(M) has in U components rc"y of the form
(
L1
01
(vIri'y)= ...'L.
o
Or-.where (2.4)
with respect to the frame ICy}, v" being an arbitrary vector field in M.
L.-L~~" - "':W"). .~
1, "', '.hxJc being, for each fixed index a. a local covector field in U. which is non-zero;
(B) For any intersecting coordinate neighborhoods U and U' of M. let ICy} and {C'y} be the frame of orthonormal cross-sections ofN(M) along U and U' respectively.
Then the matrix rp=(rpy") belongs to 1/). where C'y=rpy"C" in Un U'.
Such an orthonormal frame ICy} as constructed above is called an adapted 'IUJNIIal frame in U.
Taking aecount of
Cl.
12) and using (2. 3) and (2. 4). we see that the curvature tensor of the connection i7 has in U components [(dCY" of the form(2.5)
6 KentarO Yano and Shigeru Ishihara
with respect to an adapted normal frame {Cy } . ut' and v" being arbitrary vector :fields in M. where
(2.6)
a=l. ···.sand
(2. 7) Fw=(o},(ri).-o,)wd)dydl\dy<
is a 2·form de:6ned globally in M. for each :fixed index a. since the matrix (2. 5) with 0,. defined by (2. 6)is invariant under the group'" of the form (2.I). We may assume that each of F (ri) is a non-zero 2-form. If dim tP=d>O. then there are d members of Fw·s which are linearly independent with rel'pect to constant coefficients.
We assume. moreover. that the submanifoldMhas parallel second fundamental tensor and that the enveloping manifold is of constant curvature. Then. by Lemma 1. 2. we find f'$4./<=0. from which,
(2.8) a=l. ···.s.
Thus each of F(ei) is in M a harmonic tensor of degree 2. Hence we have
LEMMA2. 1. Let M be a S'IIbmanifold satisfying the conditions:
(P) M is immersed in a R.iemannian manifold ofCQIIStant curvature c;
(0) M has parallel secUlId frmdamental tensor;
(R) The Iwlonomy group I/> ofthe CUlInection " induced in the 1UJT'11Ia/,bundle N(M) ofM is alJelian.
If M is compact. and, ifB2=O,Bz being the secondBettinumber'ofM, then dim(11=0.
We suppose that a submanifold M satisfying the conditions (P). (Q) and CR) is irreducible. Then, taking account of (2. 8). we see that, if dim 1/»0. there is at least one member. say F. of Fw's such that F is a non·zero 2·form and kF. k being a non-zero constant, is the fundamental 2·form J of a certain Kaehlerian structure (g. J) in M. where g is the induced metric of:M. Thus we have
LEMMA2.2. Ifa submanifoldM satisfying tba cmuJitWns (P). CQ) and(R) is irre- dtu:ible. andif dimtP>O. thenthereis in M a certainKIle'hlerianstltructure(g. J). where g isthe induced metric ofM.
Next, we assume that a submanifold M satisfying the conditions (P). (0) and eR) is irreducible and that M admits no Kaeblerian structure (g. J), g being the induced metric of M. Then, by Lemma 2.2, we :find dimIt>=0. that is, the curvature tensor
K4•y" of the connection "vanishes. Thus, for each point of M, there is a coordinate
neighborhood U containing this point such that there is in U an adapted normal frame
fC
y}, withrespect to which" has vanishing components, i.e., rd"y=o. Thus each of the second fundamental tensor h••"is, for each :fixed index:.c. a symmetric tensor field of type (0.2) in U and has vanishing covariant derivative, i.e. ,f'4h••"=04h••"-{i'.Jn,.."-L".}h..."=O
because of (1.16) with Fb=O. On the other hand, the Riemannian manifold (V, g),
Submanifolds of codimension 2 or 3 wish parallel second fundamental tensor. 7 which is an open submanifold of (M, g), is irreducible, because (M, g) is irreducible and symmetric. Thus we obtain
h'bx
= g'bAX,
where A=NC is a parallel local cross-section in N(M). Summing up, we have
LEMMA 2. 3. If a submanifold M satisfying the conditions (P), (Q) and (R) is irreducible, and ifM admits no Kaehlerian structure (g, J), g being the induced metric of M, then M is umbilical or totally geodesic.
As a corollary to Lemma 2. 3, we have
LEMMA 2.4. If a submanifoldM satisfying the conditions (P), (Q) and (R) is odd- dimensional and irreducible, then M is umbilical or totally geodesic.
Let M be differentiably homeomorphic to a natural sphere. Then M admits no Kaehlerian structure, if dimM>2. On the other hand, any Riemannian manifold having a natural sphere as its underlying manifold is irreducible. Thus, from Lemma 2.3, we have
LEMMA2.5. If a submanifold M satisfying the conditions (P), (Q) and (R) is differentiably homeomorphic to a natural sphere, and if dim11-1'>2, then M is umbilical or totally geodesic.
From now on, in this section, we denote by M a 2-dimensional submanifold satisfying the conditions (P), (Q) and (R). In a coordinate neighborhood U of M, we take an adapted normal frame {Cy } , which satisfies the conditions (A) and (B). We suppose
that the 2-form
0= (odl( 1 ) , -o,l(1)d)dydI\dy'
corresponding to 01 and appearing in (2.5) is not zero. Since dimM=2, we can put in U
the function f being zero nowhere in U because of fJ,Kd,yx=O given in Lemma 1. 2. On the other hand, since dimM=2, we can put in U
(2.10) Kd,b·=A(olg,b-Oc"gdb),
A being a function inM. Since the submanifoldM satisfies the condition(Q), we have fJdh'bx=O, which implies Cl,21). The condition (1. 21) reduces, for e=l, d=2, x=
m+l. to (2. 11)
and, for e=l, d=2, x=m+2, to (2.12)
by means of (2.9) and (2.10), where we have put P'b=h'bm+l and Q'b=h'bm+2•
Now, if we take account of the fact that the mean curvature vector H=(l/n)g,bh'bXCx is parallel in N(M), we have g,bh,bm+l=O and g,bh,bm+2=0 in U. Thus, putting P'b=
It'b m+1 and Q'b=h'bm+2, we obtain gcbP'b=O and g,bQ'b=O. Thus, taking suitably in U
8 Kentaro Yano and SlUgeruIshihara an orthonormal frame {eb eal tangent to M, we have
(2.13) (P.,,)=
(a 0),
o
-al<N:::;;'m-n
with respect to
tel.
eal. Substituting (2.13) into thetensor equations (2.11) and (2.12)- with respect to {eb eal. i e.. the tensor equation (2.11) and (2.12) with g."=O.,,. we- find2aA+fc=O. fb=O and 2Ab+fa=O, 2&=0.
respectively. From these equations, we have a=b=c=O, because f is zero nowhere in U. Therefore P..=Q.,,,=O in U, that is, h.""'+l=h.,,m+2=O. Ifwe substitute h.,,"'+l=h.,,"'u
=0 into (1.19), then in U we obtain Kdcl2=O, i e., 01=0, which contradicts the assumption that the 2·form 01 is not zero in U. Consequently, the 2-form 01 necessa- rily vanishes in M. Similarly, all other 2-forms
0.:
s appearing in (2.5) vanish in M.That is to say, the curvature tensor Kd.:l' of the connection " induced in N(M) vanishes identically in M. Summing up, we have
LEMMA2. 6. Let M be a 2-dimensionalS'II1mumifold satisfyingtke conditions(P), (Q) and (R). Then the Cll3TJQture tensor ofthe CQR1I.ect;iqn If induced in N(M)is zero in M.
If M is irreducible (tWt flat with respect to tke induced metric g), then M is umbilical' or totally geot1uic.
If, in Lemma 2.6, the enveloping manifold is a Euclidean space or a sphere, and if the submanifold M is complete, connected and irreducible, then M is a 2-dimensional natural sphere. If, in Lemma 2.6, the enveloping manifold is a Euclidean space or a sphere, and if the submanifold M is complete, connected and fiat with respect to the' induced metricg, thenMisa a plane or a pythagorean product of the form81(a)xS1(b).
where $lea) denotes a natural circle of radius a.
§ 3. SubmaDifolds of eodbneusion 2
We need, in the sequel, the following theorems A and B proved in [2J (cf. [4J):
THEOREMA. Let M be a connected and complete nibmanifold of dimension n wit~.
parallel secmuJ fundamental tensor immersed in 4 Euclidean space R'" of dimension m O<n<m) andsuppose thatthe CUTTJature tensor ofthe connection induced in tke n:J17nal btmdle N(M) of M is zero. Then M is an n-dimensional sphere S"(4), n-dimensional' plane R", 4 pythagorean products ofthe form
SP,(aI)X •••xS'N(aN);
P1+"'+PN=n; PI>'" >PN>l;
or a pythagorean produce ofthe form
SP, (avX .,.xSPN(aN)xRP;
P1+"'+PN+p=n; p>o:, h>"'>PN>l; l<N:::;;'m-n,
whereS'(a) (p>l) den«es a p-dimensitJnal sphere with radius a naturally embedded..
Submanifolds of codimension 2 or 3 with parallel second fundamental tensor. 9
Sl(a) a natural circle of radius a naturally embedded and RP a p-dimensional plane.
THEOREM B. Let M be a connected and complete submanifold of dimension n with..
parallel second fundamental tensor immersed in an m-dimensional sphere S"'(a) with radius a (O<a, 1<n<m) and suppose that the curvature tensor of the connection indru:ecl in the normal bundle N(M) of M is zero. Then M is a sphere 8"(b) (O<b~a), a pythagorean product of the form
SP'(al)x··· xSPN (aN) , al2+"'+ai=a2;
h+"'+PN=n; Pl?;;"'>PN>l; N=m-n+l or a pythagorean product of the form
h+"'+PN=n; Pl~"'?;;PN~l; l<N~m-n.
Let M be a submanifold of codimension 2 in a Riemannian manifold. Then the normal bundle N(M) is of dimension 2. Thus the holonomy group (JJ of the connection induced in N(M) is a subgroup of 0(2) and hence abelian. Thus, from Lemma 2.1, Theorems A and B, we have
PROPOSITION 3.1. Let M be a submanifold of codimension 2 immersed in a Euclidean space and assume that the second fundamental tensor is parallel. If M is compact and has vanishing second Betti number B2, then M is a sphere S"(a) (n+2) or a pythagorean product of the form
SP, (al) x SP, (a2);
Pl+P2=n; Pl~3, h>l, P2+2, where dim M=n, provided that M is connected.
PROPOSITION 3.2. If, in Proposition 3.1, the em.eloping manifold is a sphere S"+2 (a).
then M is a sphere S"(b) (n+2, O<b~a), a pythagorean product of the form SP, (al) x SP, (a2) x SP, (ag), al2+a22+ag2=a2;
PI+h+pg=n; h'?;.h?;;3, Pg:::::l, Ps+2 or a pytagorean product of the form
SP, (a0 xSP, (a2), al2+a22:=;a2;
h+h=n; h>3, p2?;;1, h+2, where dim M=n, provided that M is connected.
From Lemma 2. 4, we have
PROPOSITION 3.3. Let M be an odd-dimensional submanifold of codimension 2 satisfy- ing the conditions (P) and (Q). If M is irreducible, then M is umbilical or totally geodesic.
From Lemmas 2. 5 and 2. 6, we have
PROPOSITION 3. 4. Let M be a submanifold of codimension 2 satisfying the conditions
10 Kentaro Yano and Shigeru Ihihara
(P) and (Q). If M is differentiahly lwmeonwrphic to a natural sphere, then M is umbilical or totally geodesic.
H, in Propositions 3. 3 and 3. 4, the enveloping manifold is a Euclidean space or a sphere, and if M is connected and complete, then the submanifold is a sphere naturally ,embedded.
H a submanifold M of codimension 2 satisfying the conditions (P) and (Q) is not minimal, then the mean curvature vector H is non-vanishing and parallel in N(M).
Thus the holonomy group f/J of the connection Pinduced in N(M) is of dimension 0, that is, the curvature tensor Kdcy" of P vanishes identically. Therefore, for such a submanifold M, we have the same results as those stated in Theormes A and B with m=n+2, where dim M=n.
Taking account of Lemma 1.1, we have, for a submanifold of codimension 2 satisfy- ing the conditions (P) and (Q), the same results as stated in Theorems A and B with m=n+2, where dim M=n, if the scalar curvature K of M satisfies K>n(n- 1)c, c 'being the curvature of the enveloping manifold.
§4. Submanifolds of eodimension 3-
Let there be given a submanifold M of codimension 3 satisfying the conditions (P) and (Q). H we assume that M is not a minimal submanifold, then we see that the mean -curvature vector H is non-vanishing and parallel in N(M). Thus, since N(M) is 3- dimensional, the holonomy group f/J of the connection induced in N(M) is a subgroup of0(2) and hence abelian. Thus, taking account of Lemmas 1. 1, 2.1, Theorems A .and B, we have
PROPOSITION 4. 1. Let M be a submanifold of codimnuion 3immersed in a Euclidean space and assume that M has parallel second fundamental tensor. Assume that M is Mt a minimal submanifold or that the scalar curvature K of M is non-negative. If M is .compact and has vanishing second Betti number ~, then M is a sphere Sn(a) (n=#=2), a .pythagorean product of the form
SP, (avxSP, (a2)xSP, {aa>;
Pl+1'2+PS=n; Pt>1'2>3, Pt>I, Pt=#=2 or a pythagorean product of the form
SP, (avxS/I'(a2);
Pl+1'2=n; Pl?::.3, 1'2>1, M2, -where dim M=n, provided that M is connected.
PROPOSITION 4.2. If, in ProPosition 4.1, the enveloping manifoldisa sphere S"+s Ca), .then M is a sphere sn(b) (n=#=.2, O<b:-:;;a), a pythagorean product of the form
SP. (al) x SIJ, (a2) x SP' (as) x SIJ. (a,.), a12+a22+as2+al=a2;
Pl+1'2+PS+P.=n; Pl>P2>Ps>3, P.>I, P.=#=2, .a pythagorean product of the form
Submanifolds of codimension 2 or 3 with parallel second fundamental tensor. 11
.()r a pythagorean product of the form
sp.
(al) xSP,(a2),Pl+P2=n: Pl~3, P2~1, Pd=2, where dim M=n, provided that M is connected.
We have the following Propositions 4. 3 and 4. 4 by the same devices as developed in the proofs of Propositions 3. 3 and 3.4 respectively.
PROPOSITION 4.3. Let M be an odd-dimensional submanifold of condimension 3 satisfying the conditions (P) and (Q) and assume that M is not minimol submanifold.
If M is irreducible, then M is umbilical or totally geodesic.
PROPOSITION 4. 4. Let M be a suhmanifold of codimension 3 satisfying the conditions (P) and (Q) and assume that M is not a minimal submanifold. If M is differentiahly homeomorphic to a natural sphere, then M is umbilical or totally geodesic.
If we take account of Lemma 1.1, we see that Propositions 4.3 and 4. 4 are establi- shed, replacing the assumption that M is not a minimal submanifold by another assump- tion that the scalar curvature Kof M satisfies K~n(n-I)c, dim M=n, c being the curvature of the enveloping manifold.
If, in Propositions 4.3 and 4. 4, the enveloping manifold is a Euclidean space or a sphere, and if the submanifold M is connected and complete, then M is a sphere naturally embedded.
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